Bayesian Modeling and Estimation of Linear Time-Variant Systems using Neural Networks and Gaussian Processes

The identification of Linear Time-V ariant (L TV) systems from input-output data is a fundamental yet challenging ill-posed inverse problem. This work introduces a unified Bayesian framework that models the system's impulse response, h( t,τ), as a stochastic process. We decompose the response into a posterior mean and a random fluctuation term, a formulation that provides a principled approach for quantifying uncertainty and naturally defines a new, useful system class we term Linear Time-Invariant in Expectation (L TIE). To perform inference, we leverage modern machine learning techniques, including Bayesian neural networks and Gaussian Processes, using scalable variational inference. We demonstrate through a series of experiments that our framework can robustly infer the properties of an L TI system from a single noisy observation, show superior data e fficiency compared to classical methods in a simulated ambient noise tomography problem, and successfully track a continuously varying L TV impulse response by using a structured Gaussian Process prior. This work provides a flexible and robust methodology for uncertainty-aware system identification in dynamic environments.1. Introduction Linear Time-V ariant (L TV) systems are fundamental to modeling dynamic processes in fields ranging from geophysics and communications to control theory (Kozachek et al., 2024; Lin et al., 2020). Unlike their time-invariant counterparts, an L TV system's behavior is described by an impulse response, h( t,τ), that changes over time, posing significant challenges for analysis and estimation (Kailath, 1962; Bello, 1963). The task of identifying h( t,τ) from input-output data is a severely ill-posed inverse problem, as one must infer a function of two variables from one-dimensional time series (Aubel and B olcskei, 2015). This work introduces a Bayesian framework for modeling such systems, where the inherent uncertainty and time-varying nature are captured probabilistically.